Definitions

Integral Calculus

Definite Integral

• Concept: The definite integral is the accumulated sum of the values of a function over an interval [a, b].

• Mathematical Definition:

  $$\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$

  where $\Delta x$ is the width of the subintervals, $x_i^*$ is a sample point in each interval, and the function $f(x)$ is continuous on [a, b].

Anti-Derivative

• Concept: An anti-derivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$.
• Relationship: If $F'(x) = f(x)$, then $F(x)$ is an anti-derivative of $f(x)$.

Indefinite Integral

• Concept: The indefinite integral represents a family of functions $F(x)$ that are anti-derivatives of $f(x)$.

• General Form:

  $$\int f(x) \, dx = F(x) + C$$

  where $C$ is the constant of integration, encapsulating all possible anti-derivatives of $f(x)$.